QERA: an Analytical Framework for Quantization Error Reconstruction

Cheng Zhang, Jeffrey T. H. Wong, Can Xiao, George Anthony Constantinides, Yiren Zhao

Keywords: parameter-efficient fine-tuning, PEFT, LoRA, QLoRA, quantization, post-training quantization, PTQ, low-rank approximation

TL;DR: Analytical solution to low-rank quantization error reconstruction and its applications

Abstract:

The growing number of parameters and computational demands of large language models (LLMs) present significant challenges for their efficient deployment. Recently, there is an increasing interest in quantizing weights to extremely low precision while offsetting the resulting error with low-rank, high-precision error reconstruction terms. The combination of quantization and low-rank approximation is now popular in both adapter-based, parameter-efficient fine-tuning methods such as LoftQ and low-precision inference techniques including ZeroQuant-V2. Usually, the low-rank terms are calculated via the singular value decomposition (SVD) of the weight quantization error, minimizing the Frobenius and spectral norms of the weight approximation error. Recent methods like LQ-LoRA and LQER introduced hand-crafted heuristics to minimize errors in layer outputs (activations) rather than weights, resulting improved quantization results. However, these heuristic methods lack an analytical solution to guide the design of quantization error reconstruction terms. In this paper, we revisit this problem and formulate an analytical framework, named Quantization Error Reconstruction Analysis (QERA), and offer a closed-form solution to the problem. We show QERA benefits both existing low-precision fine-tuning and inference methods -- QERA achieves a fine-tuned accuracy gain of $\Delta_{\text{acc}}$ = 6.05% of 2-bit RoBERTa-base on GLUE compared to LoftQ; and obtains $\Delta_{\text{acc}}$ = 2.97% higher post-training quantization accuracy of 4-bit Llama-3.1-70B on average than ZeroQuant-V2 and $\Delta_{\text{ppl}}$ = $-$ 0.28 lower perplexity on WikiText2 than LQER.

Primary Area: foundation or frontier models, including LLMs

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 7054